Methods Based on Rational Function Approximation of Green's Function Spectra

In this chapter, the authors describe the rational function fitting method (RFFM) and spectral differential equation approximation method (SDEAM), which shift the burden of numerical computations from the evaluation of the slowly converging and highly oscillating Sommerfeld integrals to the approximation of their integrands with rational functions, which enable use of standard identities allowing for analytic evaluation of these integrals. The vector fitting‐based RFFM requires availability of the analytic solution of the one‐dimensional boundary value problems for the spectra of vector potential components. SDEAM solutions both for the vector potential Green's function components in the traditional formulation and mixed‐potential Green's function components discussed can be performed with high‐order discontinuous Galerkin method. By analogy with planar layered medium, casting of the Green's function spectrum into the pole‐residual form allows to evaluate space domain Green's function in spherical layered medium in closed‐form as well Okhmatovski and Cangellaris.